Documentation

Init.Guard

Reducible defeq matching for guard_hyp types

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    Default-reducibility defeq matching for guard_hyp types

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      Syntactic matching for guard_hyp types

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        Alpha-eq matching for guard_hyp types

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          The guard_hyp type specifier, one of :, :~, :ₛ, :ₐ

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            Reducible defeq matching for guard_hyp values

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              Default-reducibility defeq matching for guard_hyp values

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                Syntactic matching for guard_hyp values

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                  Alpha-eq matching for guard_hyp values

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                    The guard_hyp value specifier, one of :=, :=~, :=ₛ, :=ₐ

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                      Reducible defeq matching for guard_expr

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                        Default-reducibility defeq matching for guard_expr

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                          Syntactic matching for guard_expr

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                            Alpha-eq matching for guard_expr

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                              The guard_expr matching specifier, one of =, =~, =ₛ, =ₐ

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                                Tactic to check equality of two expressions.

                                • guard_expr e = e' checks that e and e' are defeq at reducible transparency.
                                • guard_expr e =~ e' checks that e and e' are defeq at default transparency.
                                • guard_expr e =ₛ e' checks that e and e' are syntactically equal.
                                • guard_expr e =ₐ e' checks that e and e' are alpha-equivalent.

                                Both e and e' are elaborated then have their metavariables instantiated before the equality check. Their types are unified (using isDefEqGuarded) before synthetic metavariables are processed, which helps with default instance handling.

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                                  Tactic to check equality of two expressions.

                                  • guard_expr e = e' checks that e and e' are defeq at reducible transparency.
                                  • guard_expr e =~ e' checks that e and e' are defeq at default transparency.
                                  • guard_expr e =ₛ e' checks that e and e' are syntactically equal.
                                  • guard_expr e =ₐ e' checks that e and e' are alpha-equivalent.

                                  Both e and e' are elaborated then have their metavariables instantiated before the equality check. Their types are unified (using isDefEqGuarded) before synthetic metavariables are processed, which helps with default instance handling.

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                                    Tactic to check that the target agrees with a given expression.

                                    • guard_target = e checks that the target is defeq at reducible transparency to e.
                                    • guard_target =~ e checks that the target is defeq at default transparency to e.
                                    • guard_target =ₛ e checks that the target is syntactically equal to e.
                                    • guard_target =ₐ e checks that the target is alpha-equivalent to e.

                                    The term e is elaborated with the type of the goal as the expected type, which is mostly useful within conv mode.

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                                      Tactic to check that the target agrees with a given expression.

                                      • guard_target = e checks that the target is defeq at reducible transparency to e.
                                      • guard_target =~ e checks that the target is defeq at default transparency to e.
                                      • guard_target =ₛ e checks that the target is syntactically equal to e.
                                      • guard_target =ₐ e checks that the target is alpha-equivalent to e.

                                      The term e is elaborated with the type of the goal as the expected type, which is mostly useful within conv mode.

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                                        Tactic to check that a named hypothesis has a given type and/or value.

                                        • guard_hyp h : t checks the type up to reducible defeq,
                                        • guard_hyp h :~ t checks the type up to default defeq,
                                        • guard_hyp h :ₛ t checks the type up to syntactic equality,
                                        • guard_hyp h :ₐ t checks the type up to alpha equality.
                                        • guard_hyp h := v checks value up to reducible defeq,
                                        • guard_hyp h :=~ v checks value up to default defeq,
                                        • guard_hyp h :=ₛ v checks value up to syntactic equality,
                                        • guard_hyp h :=ₐ v checks the value up to alpha equality.

                                        The value v is elaborated using the type of h as the expected type.

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                                          Tactic to check that a named hypothesis has a given type and/or value.

                                          • guard_hyp h : t checks the type up to reducible defeq,
                                          • guard_hyp h :~ t checks the type up to default defeq,
                                          • guard_hyp h :ₛ t checks the type up to syntactic equality,
                                          • guard_hyp h :ₐ t checks the type up to alpha equality.
                                          • guard_hyp h := v checks value up to reducible defeq,
                                          • guard_hyp h :=~ v checks value up to default defeq,
                                          • guard_hyp h :=ₛ v checks value up to syntactic equality,
                                          • guard_hyp h :=ₐ v checks the value up to alpha equality.

                                          The value v is elaborated using the type of h as the expected type.

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                                            Command to check equality of two expressions.

                                            • #guard_expr e = e' checks that e and e' are defeq at reducible transparency.
                                            • #guard_expr e =~ e' checks that e and e' are defeq at default transparency.
                                            • #guard_expr e =ₛ e' checks that e and e' are syntactically equal.
                                            • #guard_expr e =ₐ e' checks that e and e' are alpha-equivalent.

                                            This is a command version of the guard_expr tactic.

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                                              Command to check that an expression evaluates to true.

                                              #guard e elaborates e ensuring its type is Bool then evaluates e and checks that the result is true. The term is elaborated without variables declared using variable, since these cannot be evaluated.

                                              Since this makes use of coercions, so long as a proposition p is decidable, one can write #guard p rather than #guard decide p. A consequence to this is that if there is decidable equality one can write #guard a = b. Note that this is not exactly the same as checking if a and b evaluate to the same thing since it uses the DecidableEq instance to do the evaluation.

                                              Note: this uses the untrusted evaluator, so #guard passing is not a proof that the expression equals true.

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