User module:Mattgately/Tau day2

Proving theorems from Interface:Tau_day_theorems_2
 import (PROPOSITIONS Interface:Classical_propositional_calculus ) import (FIRSTORDER Interface:First-order_logic_with_quantifiability (PROPOSITIONS) ) import (TAU Interface:Tau_day_axioms (PROPOSITIONS FIRSTORDER) )

thm(Cosine4  ((cos ((τ) / (4))) = (0)) ( ((τ) / (4)) (0) Subtraction                      # tau/4-0 = tau/4 + -0 (0) AdditiveInverse        #    0 + -0 = 0 swapEquality            #     0 = 0 + -0 (-(0)) AdditiveIdentityLeft     # 0 + -0 = -0 applyEqualityTransitivity             # 0 = -0 swapEquality                   # -0 = 0 ((τ) / (4)) buildAdditionLL                      #  tau/4 + -0 = tau/4 + 0 applyEqualityTransitivity                        # tau/4-0 = tau/4 + 0 ((τ) / (4)) AdditiveIdentity                    #     tau/4 + 0 = tau/4 applyEqualityTransitivity                        #     tau/4-0 = tau/4 buildCosine                                      #     cos tau/4-0 = cos tau/4 swapEquality                                    #    #     cos tau/4 = cos tau/4-0 (0) CosineComplement                        #   cos(tau/4-0)=sin(0) Sine0                                       #       sin(0)=0 applyEqualityTransitivity                   #      cos(tau/4-0) = 0 applyEqualityTransitivity ))

thm(Cosine8  ((cos ((τ) / (8))) = ((√ (2)) / (2))) ( (τ) EighthQuarterHalf buildCosine NegativeTau2LessEqualTau4 Tau4LessEqualTau2 introduceConjunction ((τ) / (4)) CosineHalfAngle applyModusPonens applyEqualityTransitivity       # gives      cos(tau/8) = sqrt( (1+cos tau/4)/2 ) Cosine4 (1) buildAdditionLL                   #    gives    1+cos(tau/4)=1+0 (1) AdditiveIdentity applyEqualityTransitivity               #  gives   1+cos(tau/4)=1 TwoNotZero buildDivisionRR                 #   gives   ( 1+cos(tau/4) )/2 = 1/2 buildSquareRoot                #   ((x = y)) ((√ x) = (√ y)))    SquareRootOneHalf              #    ((√ ((1) / (2))) = ((√ (2)) / (2)))) applyEqualityTransitivity          #           gives        √   ( 1+cos(tau/4) )/2     =   ((√ (2)) / (2)))) applyEqualityTransitivity      #    gives       cos(tau/8)     =   ((√ (2)) / (2)))) ))

thm(Sine8  (  ( sin ((τ)/(8)) ) = ( (√(2)) / (2) )  ) ( (τ) QuarterMinusEighth     # (((x / (4)) − (x / (8))) = (x / (8))))    swapEquality    buildSine    ((τ)/(8)) SineComplement          # ((sin (((τ) / (4)) − θ)) = (cos θ))) Cosine8                        # ((cos ((τ) / (8))) = ((√ (2)) / (2))))    applyEqualityTransitivity    applyEqualityTransitivity ))

thm(CosineNegative4  (   (  cos ( - ((τ)/(4)) )  ) = (0)   ) ( ((τ)/(4)) CosineNegation           # ((cos (- θ)) = (cos θ)))    Cosine4                            # ((cos ((τ) / (4))) = (0))) applyEqualityTransitivity ))

thm(Sine2  ((sin ((τ) / (2))) = (0)) ( ((τ) / (2)) CosineComplement           #((cos (((τ) / (4)) − θ)) = (sin θ)))    swapEquality    Tau4MinusTau2                    # ((((τ) / (4)) − ((τ) / (2))) = (- ((τ) / (4))))) buildCosine applyEqualityTransitivity CosineNegative4 applyEqualityTransitivity ))

export (THEOREMS Interface:Tau_day_theorems_2 (PROPOSITIONS FIRSTORDER) ) 